Fast MRI with sparse sampling: models, algorithms, and applications

Conventional magnetic resonance imaging (MRI) methods are based on the Shannon-Nyquist sampling theorem. The number of required Nyquist samples grows exponentially with respect to the underlying physical dimension of the imaging problem, resulting in significant difficulty of achieving high resolution for higher-dimensional imaging applications. This research addresses such a problem from a sparse sampling perspective. We have proposed novel constrained imaging approaches, including imaging models and reconstruction algorithms, to enable high-quality reconstruction from highly undersampled data. The utility of the proposed techniques is demonstrated in two challenging higher-dimensional MRI applications, i.e., dynamic MRI and MR parameter mapping. First, we propose a novel constrained image model, the joint low-rank and sparse model, to enable dynamic image reconstruction from highly undersampled (k, t)-space data. Low-rank and sparse models are two low-dimensional signal structures, each of which parsimoniously models dynamic imaging data. Here, we integrate the two models into a single mathematical formulation. With pre-estimation of the temporal subspace for the low-rank model, the proposed formulation results in a convex optimization problem, for which we develop a globally convergent algorithm based on the half-quadratic regularization with continuation procedure. We systematically demonstrate the complementary advantages of incorporating both low-rank and sparsity models into the proposed formulation with a real-time cardiac imaging application. Second, we extend the joint low-rank and sparsity model to accelerate an important class of quantitative MRI problems, i.e., MR parameter mapping. We specifically tailor the low-rank and sparse model to capture different signal/image characteristics in parameter mapping applications: the low-rank constraint captures the correlation among relaxation signals at different spatial locations, whereas the sparsity constraint captures the edge structure shared by a sequence of co-registered, contrast-weighted MR images. We demonstrate the superior performance of the proposed method over two state-of-the-art methods that are based on a single low-rank or sparse model with in-vivo data. Third, we address the MR parameter mapping problem from a different perspective with a novel model-based parameter estimation approach. Specifically, we propose a mathematical formulation that integrates an explicit parametric signal model from MR physics with sparsity constraints on the model parameters. It enables direct estimation of the parameters of interest from highly undersampled, noisy k-space data. We present an efficient greedy-pursuit algorithm to solve the resulting optimization problem. We also derive estimation-theoretic bounds to analyze the benefits of sparsity constraints and benchmark the performance of the proposed method. Both the theoretical properties and empirical performance of the proposed method are illustrated in a T2 mapping application example